University of Groningen

Non-linear model based control of a propylene polymerization reactorAl-Haj Ali, M.; Betlem, B.; Weickert, G.; Roffel, B.

Published in:Chemical Engineering and Processing

DOI:10.1016/j.cep.2006.07.012

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Citation for published version (APA):Al-Haj Ali, M., Betlem, B., Weickert, G., & Roffel, B. (2007). Non-linear model based control of a propylenepolymerization reactor. Chemical Engineering and Processing, 46(6), 554-564.https://doi.org/10.1016/j.cep.2006.07.012

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https://doi.org/10.1016/j.cep.2006.07.012https://research.rug.nl/en/publications/nonlinear-model-based-control-of-a-propylene-polymerization-reactor(b35b4a61-8dec-480e-910c-8c526e4e9b9c).htmlhttps://doi.org/10.1016/j.cep.2006.07.012

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Chemical Engineering and Processing 46 (2007) 554–564

Non-linear model based control of a propylene polymerization reactor

M. Al-Haj Ali, B. Betlem, G. Weickert, B. Roffel ∗Faculty of Science and Technology, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands

Received 2 September 2005; received in revised form 14 March 2006; accepted 20 July 2006Available online 2 August 2006

bstract

A modified generic model controller is developed and tested through a simulation study. The application involves model-based control of aropylene polymerization reactor in which the monomer conversion and melt index of the produced polymer are controlled by manipulating theeactor cooling water flow and the inlet hydrogen concentration.

Non-linear control is designed using a simplified non-linear model, in order to demonstrate the robustness of the control approach for modelingrrors. Two model parameters are updated online in order to ensure that the controlled process outputs and their predicted values track closely.he controller is the static inverse of the process model with setpoints of the measured process outputs converted to setpoints for some of the state

ariables.

The simulation study shows that the proposed controller has good setpoint tracking and disturbance rejection properties and is superior to theonventional generic model control and Smith predictor control approaches.

2006 Elsevier B.V. All rights reserved.

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eywords: Non-linear control; Model-based control; Polymerization; Non-line

. Introduction

Control of polymerization reactors is probably one of theost challenging issues in control engineering. The difficulties

n operating such processes are numerous. Firstly, the processynamics are often highly non-linear because of the compli-ated reaction mechanisms associated with the large number ofnteractive reactions. Secondly, on-line monitoring of polymeruality is often hampered by a lack of on-line measurementsor key quality variables such as composition (or monomer con-ersion), molecular weight and copolymer composition [1]. Ifeasuring quality variables is at all possible, there may still be a

umber of problems associated with these measurements, suchs (i) sampling problems, (ii) large dead times, (iii) off-line anal-sis, and (iv) sometimes large measurement errors and/or highoise levels. A more detailed discussion of measurement dif-culties in the field of polymerization can be found, amongst

thers, in Kiparssides [2]. To cope with the lack of on-lineeasurements of polymer quality, researchers have employed

ifferent inferential and estimation techniques [1,3–5].

∗ Corresponding author.E-mail address: b.roffel@utwente.nl (B. Roffel).

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255-2701/$ – see front matter © 2006 Elsevier B.V. All rights reserved.oi:10.1016/j.cep.2006.07.012

del; Model parameter update

Many articles have been published in the area of polymereactor control in the last few years. They can be divided intoinear and non-linear control approaches. There are numerousxamples in the literature of linear control approaches appliedo polymerization reactor control, such as, PI cascade control6], dynamic matrix control [7,8], generalized predictive control9] and adaptive internal model control [10]. Examples of thepplication of non-linear control approaches are, amongst oth-rs, globally linearizing control [11–13] and non-linear modelredictive control [14,15]. There are also some approaches inhich linear control is used, combined with non-linear models

or setpoint updating [16].Another type of control that has received moderate attention

s generic model control (GMC). This method uses a non-linearrocess model and assuming a desirable process output trajec-ory, a non-linear control law can be derived. A recent examplef its application in combination with extended Kalman filterings found in Arnpornwichanop et al. [17].

In the current paper an approach similar to generic modelontrol is being proposed, although its implementation and tun-

ng is simpler. It implements the non-linear model of the processirectly and gives an on-line estimation for the delayed measure-ents (Fig. 1); thus, there is no need to design an estimator, such

s a Kalman filter. This control strategy is applied to the polymer-

mailto:b.roffel@utwente.nldx.doi.org/10.1016/j.cep.2006.07.012

M.A.-H. Ali et al. / Chemical Engineering

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ig. 1. Reactor control based on simplified non-linear model, using model andontroller update.

zation of propylene in a fully-filled hollow shaft reactor [18].n case of a perfect non-linear model, a perfect non-linear con-roller can be designed. In case of a simplified non-linear model,he control system is improved by updating two model parame-ers of the simplified process and control models using an online

odel parameterization method. The efficiency of this controllgorithm is compared to the performance of a conventional PIontrol system with Smith predictor dead time compensation.

The advantages of the proposed control approach over otherpproaches are: (i) there is no need for use of an extendedalman filter to estimate unknown states or parameters, (ii) there

s no need to solve the coupled set of non-linear ordinary differ-ntial equations, and (iii) the controller shows a good robustnesshe adaptation of the model parameters, as a result of whichrrors in dynamics and kinetics can easily be dealt with.

. Non-linear control

Consider a process, which can be described by the followingquations:

dx

dt= f (x, p) + g(x, u) + l(x, d)

y = h(x)(1)

here x is the vector of state variables, y the vector of measuredariables, u the vector of input variables, d the vector of distur-ance variables, p the vector of process parameters, and h, f, g,are the non-linear function vectors.

Let the model be a simplified description of the process withdifferent parameter set p and be given by:

dx̂

dt= f (x̂, p̂) + g(x̂)u + l(x̂)d

ŷ = h(x̂)(2)

here the hat refers to the model values. In the developmentf the generic model control algorithm it is assumed that theerivative of y obeys the following equation [19]:∫ tf

dy

dt= K1(ysp − y) + K2

0(ysp − y) dt (3)

ere K1 and K2 are tuning parameters and ysp is the setpointalue of the process output. Using Eq. (2), the derivative of the

omvs

and Processing 46 (2007) 554–564 555

tate variable can be expressed as:

˙̂ = ˙̂y[

dh(x̂)

dx̂

]−1(4)

ubstitution of the derivative of x̂ in Eq. (2) results in:

˙̂[

dh(x̂)

dx̂

]−1= f (x̂, p̂) + g(x̂)u + l(x̂)d (5)

rom which the equation for the control input vector can beerived:

=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

K1(ysp − y) + K2∫ tf

0 (ysp − y) dt−(dh/dx̂)[f (x̂, p̂) + l(x̂)d]

(dh/dx̂)g(x̂)

⎫⎪⎪⎪⎬⎪⎪⎪⎭

(6)

f the model is not linear in the control vector u, its values haveo be computed through iteration. The parameters K1 and K2 areuning parameters. If the model is not perfect, control perfor-

ance will deteriorate, and the integral action in the controllerill eliminate offset. However, it is preferred to use parameter

stimation in order to update the model and thus account forarameter and structural errors. Farza et al. [20] suggested aimple non-linear observer, although other estimation schemesre possible, such as, e.g. a Kalman filter.

The tuning parameters K1 and K2 enable us to tune suchhat even some overshoot can be realized. This can primarily beealized through adjustment of K1. A disadvantage of tuning forome overshoot in one variable is that it also affects the responsef the other controlled variables. A smoother response withoutvershoot will show a smoother response of the other controlledariables.

If parameter update ensures that the model output tracks therue process output, the integral term in Eq. (6) is not required,ince there will be no sustained offset in the controlled variables.ence if K2 = 0 and tuning of K1 is done very conservatively to

uppress variable interaction, one may wonder why one wouldot use a controller with both tuning values K1 and K2 set equalo zero, i.e. use a controller that is based on a static process modelith parameter update. This may give a conservative response

or setpoint changes, which approaches the open loop responsef the system, however, disturbance rejection properties arexpected to be good. The controller can then be calculated byhe following set of equations:

= −f (x̂sp, p̂) − l(x̂sp, d)g(x̂sp)

, x̂sp = h∗(ysp, x̂) (7)

here the estimated setpoint values of the output vector coulde filtered values of the true setpoint values and the parameter

ˆ needs to be updated. In Eq. (7) the dimension of the y vectors usually smaller than the dimension of the x vector, thereforeot all state variables setpoint values can be calculated, conse-uently, some setpoint values are set equal to the current values

f the state variables from the model. This is also one of theain differences with generic model control where all the state

ariables follow from the process model and none of them haveetpoint targets.

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56 M.A.-H. Ali et al. / Chemical Engine

The parameter update should be realized such that the pre-icted process output values do track the true process measure-ents. Assume that it is required that the predicted process

utput values follow the true process outputs according to arst order response with time constant τ:

dŷ

dt= y − ŷ (8)

n a steady state situation, when u and d are constant, an offsetetween y and ŷ can only be minimized through adjustment ofhe model parameter p̂. Eq. (8) can then be rewritten as:

dp̂

dt= 1

τ

y − ŷ∂ŷ/∂p̂

(9)

n some cases it may be easier to rewrite Eq. (9) in a some-hat different manner. In that case, Eq. (8) is differentiated with

espect to p̂ and substituting back into Eq. (9), which gives:

dp̂

dt= − 1

τ2

y − ŷ∂[dŷ/dt]/∂p̂

(10)

. Process description

The hollow shaft reactor is an experimental extruder-like con-inuous reactor with internal recycling of the reaction medium.t has been designed for polymerizations at high viscosities upo a few hundred Pa s, and to work under high pressure andemperature, 250 bar and 250 ◦C. The reactor possesses the fol-owing properties: minimum dead volume, maximum recycleatio, fast and predictable macro mixing; the recycle ratio andacro-mixing do not depend on the viscosity of the reactionass in a wide range of viscosities [18].The reactor is used for liquid-pool propylene polymeriza-

ion with a multi-site heterogeneous Ziegler–Natta catalyst. Thenlet flow to the reactor consists of pure monomer, catalyst andydrogen, the latter is used as a transfer agent to provide a bet-er control of the molecular weight of the produced polymer. Aoolant removes the heat released due to polymerization.

One of the first considerations in establishing a control strat-gy is to arrange the system inputs and outputs into manipulated,ontrolled and disturbance variables. The polymerization pro-ess studied in this work has five inputs (manipulated and dis-urbance variables) and four controlled variables. Assuming fastooling water dynamics, input variables include cooling waterow (Fw), outlet liquid flow rate (F) and feed rate of monomerFinym,in), hydrogen (FinyH2,in), and catalyst (Finycat,in). Reactorressure (P), polymer melt index (MI), reaction conversion (C)nd temperature (T) could be used as controlled outputs.

The reactor system is equipped with an automatic valve athe outlet that controls the reactor pressure P by manipulatinghe outlet flow F. In a pilot setup, it is aimed to keep the catalystnd monomer feed rates constant. Consequently, they will note used in designing the control system. The control probleman therefore be simplified to a system with two manipulatednputs FinyH2,in and Fw, and two controlled variables MI and C.

atc

(r(

and Processing 46 (2007) 554–564

. Dynamic process model

The process model consists of dynamic material balances, aynamic energy balance and algebraic equations for kinetic ratexpressions and physical properties, as described in Appendix. The mechanism of propylene polymerization is explained

lsewhere [21]. In order not to complicate the model descriptionoo much, a number of assumptions were made, also listed inppendix A.The measurements of the process outputs, i.e. the monomer

onversion and polymer melt index are subject to measurementelays, the delay for the conversion is 1 h and for the melt indext is 2 h.

The detailed model as described in Appendix A is used ashe process description. If the model used for prediction of theontrolled output is the same as this set of equations, a perfectrediction is obtained and the non-linear controller is a perfecton-linear controller.

. Model simplification

In order to demonstrate the robustness of the control approacho modeling errors, the following deliberate simplifications werentroduced. The rate of reaction, Eq. (A.9), is approximated by:

p = k1K3mycρ̄mX (11)

here K3 is a tunable parameter with an initial value of 0.91,¯m the constant value for the monomer density and a 5% errorn the calculation of k1 is introduced.

Since the density is assumed constant, the equation for theutlet flow, Eq. (A.13), can be simplified to:

=(

Fin

ρm+ Rp

(1

ρp− 1

ρm

))ρ (12)

q. (A.19) was approximated by a linear first order differentialquation:

.9dMIc

dt= MIi − MIc (13)

nd the exponent in Eq. (A.16) was assumed to have a value equalo one. Another tunable parameter K4 was therefore introducedn Eq. (A.16) to compensate for structural and parametric model

ismatch:

Ii = K4κX (14)

ith the initial value of K4 equal to 0.88.If the inaccuracies in the model are not known, a sensitivity

nalysis should be performed to find out which equations havehe largest impact on the controlled variables in order to be aandidate for introduction of the parameter update.

Summarizing, the simplified model consists of Eqs.A.1)–(A.20), with Eq. (A.9) replaced by Eq. (11), Eq. (A.13)eplaced by Eq. (12), Eq. (A.16) replaced by Eq. (14) and Eq.A.19) replaced by Eq. (13).

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M.A.-H. Ali et al. / Chemical Engine

. Parameterization of the simplified model

The implementation of a simplified model in non-linearontroller design will usually result in unacceptable perfor-ance, the main problem being offset in the controlled vari-

bles. Thus, the parameters in the simplified model shoulde updated for prediction and control to be effective. Dif-erent updating approaches can be used. McAuley and Mac-regor [1] implemented the recursive prediction error method

or updating a set of parameters in the instantaneous meltndex and density correlations. Because of its flexibility, otheresearchers [6,22,1] preferred to use extended Kalman filteringr other types of observers such as the Luenberger estima-or [23]. Rhinehart and Riggs [24] used Newton’s method andn intuitive relaxation method to calculate the model parame-er update, our proposed method shows some resemblance tohis method. In our case we do not use relaxation as a tuningarameter, instead, we propose to use a first-order time con-

tant, which will be more acceptable from an engineering pointf view.

Fig. 2. Response to step changes in melt-index and conversion setpoints.

X

Fc

and Processing 46 (2007) 554–564 557

As shown in Appendix B, update of the model parameters K3nd K4 proceeds according to the following equation:

j,k+1 = Kj,k + αpv,kepv,k (15)

n which k is the time step, j = 3 when the process variable pvs the conversion and j = 4 when the process variable is the meltndex; e is the error between the measured process output andhe estimated process output using the simplified model. Theoefficient α depends on the process conditions.

. Non-linear controller design

Starting point for the controller design is the static simplifiedodel. The setpoint for the melt index MIc,sp can be written assetpoint for ratio of hydrogen to monomer concentration by

sing Eq. (14):

sp = MIc,spκK4

(16)

ig. 3. Response of manipulated variable to step changes in melt-index andonversion setpoints.

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Bt

y

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T

atflp

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58 M.A.-H. Ali et al. / Chemical Engine

he setpoint for the conversion can be written as a setpoint forhe monomer concentration by using Eq. (A.20):

m,sp = ym,in(1 − Csp) (17)y combining Eqs. (A.2)–(A.6), the inlet hydrogen concentra-

ion can be written as:

H2,in = yH2,sp +RH2

Fin(18)

n which RH2 follows from the simplified model equations andin is a measured variable. Eq. (A.12) can be used to calculate

he specific heats of the reactor inlet flow and the fluid inside theeactor, using the reactor temperature from the simplified model.he static version of the reactor energy balance, Eq. (A.11) canubsequently be used to calculate the reactor jacket temperature:

j,sp = Tm + 1UA

[Fin(Cp,inTin − CpTm) − Rp �HR,p] (19)

fter which the linear relationship between the jacket tempera-ure and cooling water flow can be used to compute the waterow through the reactor jacket. Tm represents the reactor tem-erature from the simplified model.

Fig. 4. Update of model parameters during setpoint changes.

tupt

and Processing 46 (2007) 554–564

The control law of Eqs. (18) and (19) is rather similar tohe one that can be derived for generic model control, how-ver, there are two main differences: (i) this controller does notave proportional integral control action to ensure that the pro-ess output follows a prescribed trajectory. In this case modelpdating ensures that there will be no process-model mismatchnd the process output approaches setpoint; (ii) the setpoints forhe controlled process outputs are converted to setpoints for theame number of state variables. This can easily be achieved,ince in reactor modeling component concentrations and tem-eratures are often measured and they are also the state variablesf the model. The control approach as described in this sections therefore called mGMC, modified generic model control.

. Conventional proportional-integral control with deadime compensation

In many polymer producing companies, classical control

echniques such as proportional integral (PI) control is still beingsed, the designed non-linear controller will therefore be com-ared to a conventional PI controller with Smith predictor deadime compensation. Using the relative gain array method (RGA)

Fig. 5. Controlled variable responses to a +20% disturbance in feed rate.

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M.A.-H. Ali et al. / Chemical Engine

25], it was found that the melt index can be best controlled byhe inlet hydrogen concentration, yH2,in, and the conversion byhe jacket temperature, Tj (i.e. cooling water flow). Due to theresence of measurement dead times of 1 and 2 h for conversionnd melt index, respectively.

. Results and discussion

Since the process model is represented by a simplified model,he responses will show process/model mismatch. As a result, thearameter update scheme will come into effect to ensure that theodel output tracks the process output. It should be mentioned

hat the update scheme uses fixed values of αC,k in Eq. (A.4) andMI,k in Eq. (A.7), equal to 0.6 and 0.001, respectively, sincehanges in these values were found to be limited to a maximumhange of 20%.

.1. Performance of the non-linear and PI controllgorithms

In this section, the performance of the following controlpproaches will be discussed: (i) the generic model controller,ii) the modified generic model controller and (iii) the PI–Smith

ig. 6. Manipulated variable responses to a 20% disturbance in feed rate.

ftug

F

and Processing 46 (2007) 554–564 559

redictor controller. Performance is examined for four differentases:

polymer grade change,conversion setpoint change,disturbance rejection,error in dead time of the melt index measurement of 1 h andconversion measurement of 0.5 h.

In the closed-loop simulations, it is assumed that the val-es of MIc and C are available every 2 and 1 h, respectively;ithin these time intervals, estimated values are obtained using

he property models and the parameter-updating scheme. Theontroller algorithms are executed every 6 min.

In addition to monitoring the controlled variables, also theanipulated variable moves are monitored.Figs. 2 and 3 show the closed-loop responses of the controlled

nd manipulated variables for a change in melt index setpointrom 15 to 30 at time t = 5 h and a change in conversion setpoint

rom 0.18 to 0.22 at time t = 35 h. Fig. 2 shows the response ofhe controlled outputs, Fig. 3 shows the responses of the manip-lated variables for completeness. Controller tuning settings areiven in Table 1. In Fig. 2 it can be seen that the generic model

ig. 7. Controlled variable responses to a 20% disturbance in catalyst activity.

560 M.A.-H. Ali et al. / Chemical Engineering

Table 1Controller tuning for setpoint changes

PI/SP conversion controller Kc = 2, Ti = 2.0PI/SP melt index controller Kc = 0.00125, Ti = 4.0MC

ciitbr

vovfsoti

F

rmttcoamp

iFToonfor the mGMC and SP/PI controller are somewhat larger thanfor the GMC controller in case of the melt index. The responseof the conversion to this change is very much the same for all

elt index setpoint filter τ = 0.2 honversion setpoint filter τ = 0.1 h

ontroller gives a rather large overshoot for a setpoint changen the cumulative melt index, at the same time the interactions visible in the conversion response (around t = 10 h). Detuninghe GMC controller improves the response, since the interactionetween the variables is reduced but also reduces the speed ofesponse.

On the one hand the GMC controller decouples the processariables through the static inverse of the process model, on thether hand a PI controller is added which introduces processariable interaction. The Smith predictor controller also suffersrom the interaction between the process variables, detuninglows down the response. As can be seen, the mGMC controller

utperforms the other two controllers. Fig. 4 shows the parame-er update during these transients. As can be seen, the responses smooth.

ig. 8. Manipulated variable responses to a 20% disturbance in catalyst activity.

c

Fd

and Processing 46 (2007) 554–564

Another issue that should be considered is load or disturbanceejection. The first type of disturbance that will be considered is aeasurable change in the propylene inlet flow rate; the rejection

ests were conducted with a 20% increase in inlet flow rate at= 5 h, retaining the controller settings for setpoint changes. Asan be seen from Fig. 5, also in this case the mGMC controllerutperforms the other two controllers. The melt index is notffected much by the disturbance, the conversion suffers from aomentary decrease (at t = 6 h) which is the largest for the Smith

redictor controller (Fig. 6).Another type of unmeasurable disturbance that is considered

s a 20% change in the catalyst activity. As can be seen fromig. 7, the GMC controller outperforms the other two controllers.his is due to the aggressive tuning of the integral action in casef GMC control, this also causes the response to be slightly morescillatory than the other two responses. All controllers reach aew steady state around the same time, the maximum deviation

ontrollers (Fig. 8).

ig. 9. Controlled variable responses to setpoint changes in case of dead timeiscrepancy.

M.A.-H. Ali et al. / Chemical Engineering

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ig. 10. Manipulated variable responses to setpoint changes in case of dead timeiscrepancy.

Another type of disturbance that could occur is a discrepancyn sampling times of the controlled outputs from the model androm the plant. The dead time of the melt index is assumed toe 2 h, the dead time of the conversion is assumed to be 1 h.igs. 9 and 10 show the impact of a discrepancy in dead timeetween the process and the model, the dead time of the processelt index is decreased by 1.0 h and the dead time of the process

onversion is decreased by 0.5 h. All controllers are affected,he mGMC controller performed better than the other two con-rollers.

0. Conclusions

Non-linear process model based control was studied forontrol of liquid propylene polymerization under varying con-itions. The controller manipulated the hydrogen flow ratend cooling water flow to follow the setpoints for cumula-

ive melt index and reaction conversion and to remove theffects of various process disturbances. The non-linear con-rol strategy was called modified generic model control, itsed the static inverse of the process model with setpoints

and Processing 46 (2007) 554–564 561

f the measured process outputs converted into setpointsor the state variables. In addition, model parameters werepdated to ensure good setpoint tracking and disturbance rejec-ion. Tuning of the proposed control strategy is simple, theime constant of the setpoint filters can be adjusted and thepeed at which the parameter update is accomplished can beelected.

Performance of the control strategy was compared to aeneric model controller and a proportional integral controllerith Smith predictor dead time compensation.Closed loop simulations revealed that for setpoint changes

he modified generic model controller was superior to the otherwo controllers, also for measurable feed disturbances it outper-ormed the other control approaches.

For unmeasurable disturbances in the catalyst activity, theesponse of the melt index was somewhat faster for the genericodel controller due to aggressive tuning of the integral action,

his also lead to a more oscillatory response.

ppendix A. Dynamic model of the polymerizationrocess

In order to develop a model of limited complexity, the fol-owing assumptions were made:

The polymerization reactions are irreversible and first orderwith respect to each reactant.The reactor is ideally mixed. Thus, no temperature and con-centration gradients are present. If the stirrer speed in thereactor is in the range of 100 rpm, the reactor is (macro) mixedwithin 40–80 s, meanwhile, the reactor average residence timemay reach 1 h.The reactor is fully filled, no gas phase is present in the reactor.The energy produced due to mixer rotation is negligible.The catalyst decay through different chemical mechanisms atvarious types of active sites may be lumped together into asingle deactivation. In addition, the active site concentrationdecreases in accordance with a first order decay mechanismconstant [26].Monomer equilibrium concentration near the active sites isassumed the same as the monomer bulk concentration. Thus,it can be calculated using a monomer density correlation.The reactor contains two phases: (i) a liquid monomer phaseand (ii) a polymer phase. The liquid phase consists of propy-lene monomer with dissolved hydrogen and the polymerphase consists of crystalline polymer and amorphous poly-mer, which is swollen with the monomer.

In this model, all variables should have a hat in order to showhey are model values, however, it has been omitted for reasonsf simplicity of notation.

The overall mass balance of the reactor can be described as:

dm

dt= Fin − F (A.1)

5 ering and Processing 46 (2007) 554–564

wfl

m

yRc

m

iirras

R

wpb

P

wtTd

q

wT

bc

m

wtb

m

T

id

Table 2Thermodynamic and physical parameters for propylene polymerization

Parameter Value

Physical parametersReactor volume (V) 1.86 × 10−3 m3Reactor heat transfer area (A) 0.0961 m2

Thermodynamic parametersOverall heat transfer coefficient (U) 1.62 MJ/h K m2

Heat of propagation reaction (�HR,p) 2.03 MJ/kgSpecific heat of polypropylene (Cp,p) 2.25 × 10−3 MJ/kg KDensity of polypropylene (ρp) 900 kg/m3

Specific heat of propylene (Cp,m)a 2.785 × 10−3 MJ/kg Kb −9.18 × 10−6 MJ/kg K2c 2.93 × 10−8 MJ/kg K3

Density of propylene monomer (ρm)ρm,a −263.7 kg/m3ρm,b 6.827 kg/K m3

ρm,c −0.0143 kg/K2 m3

Parameters for Eq. (A.16)κ 6818.3γ 1.03

Parameters for Eq. (A.17)β −2.34for X < 0.00144

d 5.32 × 10−5e 0.115

elsed 1.52 × 10−4e 0.0405

K0 in Eq. (A.10) [−204256.61, 1153.3314,−1.626207]

kd0 3746 h−1kd1 1.748 × 10−7 h−1Ea1/R 1620.8 K

Smwe

(

wjflb

c

T

62 M.A.-H. Ali et al. / Chemical Engine

here m is the total mass inside reactor and F the outlet massow rate in kg/h, Fin = 1.0 kg/h. The monomer mass balance is:

dymdt

= Fin(ym,in − ym) − Rp (A.2)

m is the mass fraction of monomer in the outlet flow stream, andp is the propagation reaction rate. The hydrogen mass balancean be described as:

dyH2dt

= Fin(yH2,in − yH2 ) − RH2 (A.3)

n which yH2 is the hydrogen mass fraction in g H2/kg materialnside the reactor and RH2 is the apparent hydrogen consumptionate. An apparent consumption rate is used, since the reactionate constants for the hydrogen reactions, transfer with hydrogennd dormant sites reactivation, are not known for the catalystystem used in this work.

The hydrogen consumption rate, RH2 , can be calculated from:

H2 =2Rp

42.1Pn(A.4)

here 2 and 42.1 are the molecular weight for hydrogen andropylene, respectively, Rp is the polymerization rate. The num-er average degree of polymerization Pn can be calculated from:

n = 2qPD

(A.5)

here PD is the polydispersity of the produced polymer, forhe catalyst used in this study it has an average value of 6.8.he polymerization termination probability q is experimentallyetermined from [21]:

= d + eX, X = 0.02104yH2ym

(A.6)

here X is the molar ratio of hydrogen to monomer in the reactor.he values of d and e are given in Table 2.

Based on the assumption that the catalyst is being activatedefore injecting it, the mass balance for the active catalyst, yc,an be described as:

dycdt

= Fin(yc,in − yc) − Rd (A.7)

here Rd is the deactivation reaction rate. The concentration ofhe deactivated catalyst, yd, can be calculated from the followingalance:

dyddt

= Fin(yd,in − yd) + Rd (A.8)

he reaction rates are calculated using the following equations:

Rp = k1mycρmXRd = kdmyc

(A.9)

n which k1 and kd are rate constants and ρm is the monomer

ensity. For the rate constants the following equations hold:

k1 = K01 + K02T + K03T 2kd = kd0e−Ea1/RT + kd1e−Ea2/RT (1 − e−Ea3/X)

(A.10)

tv

c

Ea2/R 5570.7 KEa3 498.9

ince the mass of the reactor wall is not small compared to theass of the reactor contents, the heat capacities of the reactorall and reactor contents are lumped together in the reactor

nergy balance. This balance can be written as:

mcp + mscp,s) dTdt

= Fin(cp,inTin − cpT ) − Rp �HR,p−UA(T − Tjacket) (A.11)

here the subscript ‘s’ refers to steel. The dependence of theacket temperature on the cooling water flow can be calculatedrom a static energy balance and is approximated by a simpleinear relationship. The specific heat of the reactor contents cane given by:

p = ym(a + bT + cT 2) + Cp,pyp (A.12)he values of the coefficients are summarized in Table 2. Cp,p is

he heat capacity of the polymer, it is assumed to have a constantalue.

Since the reactor is completely filled and there is a significanthange in density because the low-density monomer is converted

ering

tvc

F

ww

ρ

Ttw

ρ

Turtedt

M

Ttmttf

M

ttTbM

wEaf

wc

i

C

A

tTs

U

Sddato

e

At

K

Tr(

Ipfte

e

T(

K

The partial derivative of the cumulative melt index-time deriva-tive with respect to the adjustable parameter K4 is calculatedby substituting Eq. (14) into Eq. (A.19) and differentiation with

M.A.-H. Ali et al. / Chemical Engine

o the high-density polymer, the reactor outlet flow rate, F, willary. It can be shown that the following equation can be used toalculate this flow [27]:

=(

Fin

ρm+ Rp

(1

ρp− 1

ρm

)− mym dT

dt

1

ρ2m

dρmdT

)ρ (A.13)

here ρp is the polymer density and ρm is the monomer density,hich can be calculated from:

m = −ρm,a + ρm,bT − ρm,cT 2 (A.14)he constants of this equation are summarized in Table 2. ρ is

he density of the reaction mixture inside the reactor, it can beritten as:

= ρmρpymρp + ypρm (A.15)

he easily available measurements of the melt index are oftentilized to control the polymer quality in a homo-polymerizationeactor. In polyolefin production plants, it is well-known thathe concentration ratio of hydrogen to monomer, X, has a strongffect on the instantaneous melt index MIi. In the literature [1,28]ifferent relationships have been proposed to relate MIi to X. Inhis work, the following relationship is used:

Ii = κXγ (A.16)he numerical values of κ and γ are obtained from experimen-

al work and are listed in Table 2. Because the direct on-lineeasurement of the instantaneous polymer molecular proper-

ies is not practically realizable, the melt index is correlated tohe polymer average molecular weight (Mw). In this study theollowing semi-empirical equation is employed [29]:

Ic = αM̄βw (A.17)he values of α and β were calculated by fitting MI measurementso the off-line measurements of Mw, the values are presented inable 2. To calculate the cumulative melt index, the differentialalance for the cumulative weight average molecular weight,¯ w, is employed [30]:

dM̄wdt

= 1mp

(yp,inFin[M̄w,in − M̄w] + Rp[Mw − M̄w]) (A.18)

here mp is the mass of polymer inside the reactor. Substitutingq. (A.17) into Eq. (A.18), in addition to the assumption offree-polypropylene inlet stream, i.e. yp,in = 0, results in the

ollowing differential equation for the cumulative melt index:

m(1 − ym)βRp

d

dtMIc = [MI1/βi MI1−(1/β)c − MIc] (A.19)

hich is a first-order relationship with variable gain and timeonstant.

To complete the model description, the monomer conversions calculated from:

= 1 − ymym,t=0

(A.20)

r

and Processing 46 (2007) 554–564 563

ppendix B. Parameterization of the simplified model

To parameterize the conversion model, a relationship betweenhe conversion and the model parameter K3 has to be derived.his relationship can be obtained by differentiating the conver-ion equation, Eq. (A.20) with respect to time:

dC

dt= − 1

ym,t=0dymdt

(B.1)

sing Eqs. (A.2) and (11), this equation can be rewritten as:

dC

dt= −1

ym,inm[Fin(ym,in − ym) − k1K3mycρ̄mX] (B.2)

ince the relationship between C and K3 is represented by aifferential equation, Eq. (10) is used for parameter update. Theifference between the conversion using the plant measurementnd the estimated conversion using the simplified model is usedo make incremental adjustments to K3 at each execution intervalf the discrete controller:

C,k = Cplant,k − Cmodel,k (B.3)fter a simple Euler discretization of Eq. (10), the updating of

he parameter K3 can be evaluated according to:

3,k+1 = K3,k + �teC,kτ2(∂(dC/dt)/∂K3)

= K3,k + αC,keC,k(B.4)

he partial derivative of the conversion-time derivative withespect to the adjustable parameter K3 is calculated using Eq.B.2):

∂(dC/dt)

∂K3= k1ycρ̄mX

ym,in(B.5)

n the melt index model, Eq. (14) the parameter K4 is a tunablearameter. The difference between the cumulative melt indexrom the plant measurement and the simplified model is usedo make the corrections to the model parameter every controllerxecution interval:

MI,k = MIc,plant,k − MIc,model,k (B.6)he value of K4 is updated using the discretized version of Eq.

10):

4,k+1 = K4,k + �teMI,kτ2(∂(dMIc/dt)/∂K4)

= K4,k + αMI,keMI,k (B.7)

espect to K4:

∂(dMIc/dt)

∂K4= Rp

mK4(1 − ym) MI1/βi MI

1−(1/β)c (B.8)

5 ering

A

ACCFF�

kkmMMMPRR

RtTTUVXyyy

yy

Gρ

ρ

R

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

64 M.A.-H. Ali et al. / Chemical Engine

ppendix C. Nomenclature

area of heat transfer (m2)reaction conversion

p heat capacity of monomer (kJ/kg K)outlet flow rate from the reactor (kg/h)

in inlet flow rate to the reactor (kg/h)HR,p heat of polymerization (MJ/kg)

d deactivation constant (1/h)p propagation constant (m3/gcat h)

total mass inside the reactor (kg)Ii instantaneous melt index (g/10 min)Ic cumulative melt index (g/10 min)w weight average molecular weight (kg/kmol)

n number average degree of polymerizationd catalyst deactivation reaction rate (kg/m3 h)H2 average hydrogen reaction rate (m

3/gcat h)p propagation reaction rate (kg/h)

process time (h)reactor temperature (K)

j jacket temperature (K)heat transfer constant (MJ/h m2 K)reactor volume (m3)hydrogen molar ratio (mol H2/mol)

c active catalyst mass fraction (g/kg)d deactivated catalyst concentration (g/kg)H2 hydrogen mass fraction (g/kg)m monomer mass fraction in the reactor (kg/kg)p polymer mass fraction in the reactor (kg/kg)

reek symbolsdensity of reaction mixture (kg/m3)

m monomer density (kg/m3)

eferences

[1] K.B. McAuley, J.F. MacGregor, On-line inference polymer properties inan industrial polyethylene reactor, AIChE J. 37 (6) (1991) 825–835.

[2] C. Kiparssides, Polymerization reactor modeling: a review of recent devel-opments and future directions, Chem. Eng. Sci. 51 (10) (1996) 1637–1659.

[3] V. Prasad, M. Schley, L.P. Russo, B.W. Bequette, Product property andproduction rate control of styrene polymerization, J. Process Control 12(2002) 353–372.

[4] K.M. Rajendra, R.C. William, A. Penlidis, On-line non-linear model-basedestimation and control of a polymer reactor, AIChE J. 43 (11) (1997)3042–3058.

[5] H. Schuler, C.U. Schmidt, Calorimetric-state estimators for chemical reac-tors diagnosis and control—review of methods and applications, Chem.Eng. Sci. 47 (4) (1992) 899–915.

[6] T.J. Crowley, H.K. Choi, On-line monitoring and control of a batch poly-merization reactor, J. Process Control 6 (2–3) (1996) 119–127.

[7] F. Gobin, L.C. Zullo, J.P. Calvet, Model predictive control of an open-loopunstable train of polymerization reactors, Comput. Chem. Eng. 18 (Suppl.)(1994) S525–S528.

[

and Processing 46 (2007) 554–564

[8] A.M. Meziou, P. Deshpande, C. Cozewith, N.I. Silverman, W.G. Morrison,Dynamic matrix control of an ethylene-propylene-diene polymerizationreactor, Ind. Eng. Chem. Res. 35 (1996) 164–168.

[9] G. Ozkan, H. Hapoglu, M. Alpbaz, Generalized predictive control of opti-mal temperature profiles in a polystyrene polymerization reactor, Chem.Eng. Process. 37 (1998) 125–139.

10] G. Mourue, D. Dochain, V. Wertz, P. Cescamps, Identification and con-trol of an industrial polymerization reactor, Control Eng. Pract. 12 (2004)909–915.

11] K.B. McAuley, J.F. MacGregor, Non-linear product property controlin industrial gas-phase polyethylene reactors, AIChE J. 39 (5) (1993)855–866.

12] M. Shahrokhi, M.A. Fanaei, Non-linear temperature control of a batchsuspension polymerization reactor, Polym. Eng. Sci. 42 (6) (2002)1296–1308.

13] M. Soroush, C. Kravaris, Non-linear control of a batch polymerizationreactor: an experimental study, AIChE J. 38 (9) (1992) 1429–1448.

14] E.M. Ali, A.E. Abasaeed, S.M. Al-Zahrani, Optimization and control ofindustrial gas-phase ethylene polymerization reactors, Ind. Eng. Chem.Res. 37 (1998) 3414–3423.

15] G. Ozkan, S. Ozen, S. Erdogan, H. Hapoglu, M. Alpbaz, Non-linear controlof polymerization reactor, Comput. Chem. Eng. 25 (2001) 757–763.

16] R. Bindlish, J.B. Rawlings, Target linearization and model predictive con-trol of polymerization processes, AIChE J. 49 (11) (2003) 2885–2899.

17] A. Arnpornwichanop, P. Kittisupakorn, I.M. Mujtaba, On-line dynamicoptimization and control strategy for improving the performance of batchreactors, Chem. Eng. Process. 44 (2005) 101–114.

18] G. Weickert, Hollow shaft reactor: a useful tool for bulk polymerization athigh viscosities, temperatures, and polymerization rates, Ind. Eng. Chem.Res. 37 (1998) 799–806.

19] P.L. Lee, G.R. Sullivan, Generic model control (GMC), Comput. Chem.Eng. 12 (6) (1988) 573–580.

20] M. Farza, K. Busawon, H. Hammour, Simple non-linear observers foron-line estimation of kinetic rates in bioreactors, Automatica 34 (1998)301.

21] M.A.-H. Ali, B.H.L. Betlem, B. Roffel, G. Weickert, Hydrogen responsein liquid propylene polymerizations: towards a generalized model, AIChEJ52 (5) (2006) 1866–1876.

22] C. Kiparssides, P. Seferlis, G. Mourikas, A.J. Morris, Online optimizationcontrol of molecular weight properties in batch free-radical polymerizationreactors, Ind. Eng. Chem. Res. 41 (2002) 6120–6131.

23] G. Fevotte, I. Barudio, J. Guillot, An adaptive inferential measurementstrategy for on-line monitoring conversion in polymerization processes,Thermochim. Acta 289 (1996) 223–242.

24] R.R. Rhinehart, J.B. Riggs, Two simple methods for on-line incrementalmodel arrange, Comput. Chem. Eng. 15 (3) (1991) 181–189.

25] B.A. Ogunnaike, W.H. Ray, Process Dynamics, Modeling and Control,Oxford University Press, New York, 1994.

26] J.J.C. Samson, G. Weickert, A. Heerze, K.R. Westerterp, Liquid-phasepolymerization of propylene with a highly active catalyst, AIChE J. 44(1998) 1424–1437.

27] E. ten Brink, Hybrid Fuzzy-first Principles Modeling of PolypropyleneProduction in a Hollow Shaft Reactor, Chemical Technology, Enschede,University of Twente, 2002.

28] M. Ohshima, M. Tanigaki, Quality control of polymer production process,J. Process Control (10) (2000) 135–148.

29] C. Chatzidoukas, J.D. Perkins, E.N. Pistikopoulos, C. Kiparssides, Optimal

grade transition and selection of closed-loop controllers in a gas-phaseolefin polymerization fluidized bed reactor, Chem. Eng. Sci. 58 (2003)3643–3658.

30] G. Weickert, Modellierung von Polmerisationsreaktoren, Springer, Berlin,1997.

Non-linear model based control of a propylene polymerization reactorIntroductionNon-linear controlProcess descriptionDynamic process modelModel simplificationParameterization of the simplified modelNon-linear controller designConventional proportional-integral control with dead time compensationResults and discussionPerformance of the non-linear and PI control algorithms

ConclusionsDynamic model of the polymerization processParameterization of the simplified modelNomenclatureReferences